CONSTRUCTIONS OF PARTIAL DIFFERENCE SETS AND
RELATIVE DIFFERENCE SETS ON /7-GROUPS
KA HIN LEUNG AND SIU LUN MA
ABSTRACT
By using some finite local rings, we construct some new partial difference sets and relative difference sets
on ^-groups where p is any prime. When p = 2, some of the partial difference sets constructed are reversible
difference sets which include Dillon's difference sets.
1. Introduction
Let G be a finite additive group of order v. A ^-element subset D of G is called a
(v, k, X, /i)-partial difference set in G if the expressions gx—g2, for gx and g2 in D with
Si ^ #2> represent each nonzero element contained in D exactly X times and represent
each nonzero element not contained in D exactly /J, times. (Readers are reminded that
the parameters defined above are different from those in [5, 6]. A (v,k,a,/?)-partial
difference set defined in [5, 6] is, according to our definition, a (v, k, <x+/?,a)-partial
difference set, that is, X = a+/?and // = a.) It is well-known that partial difference sets
are closely related to Schur rings and strongly regular graphs (see [6]).
Suppose that v = mn and that G contains a subgroup N of order n. A ^-element
subset E of G is called an (m, n, k, X)-relative difference set in G relative to N if the
expressions gx —g2, for gx and g2 in E with gx ^ g2, represent each element contained
in G\iV exactly X times and represent no nonzero element contained in N. For related
topics of relative difference sets, see [1, 4].
In this paper, we generalize the construction of Dillon's difference sets on 2groups [3] to obtain a new construction of partial difference sets on /^-groups where
p is any prime. For p = 2, this construction give us some reversible difference sets
which include Dillon's difference sets as a special case. Finally, by using several.copies
of the partial difference sets constructed, we obtain some new relative difference sets.
The following characterizations of abelian partial difference sets and relative
difference sets are useful in the discussion later.
LEMMA 1.1. Let G be an abelian group of order v and D be a subset of G with
— D = D, that is, D is reversible. Then D is a (v, k, X, ^-partial difference set if and only
if, for any character / of G,
[k
E X(g) = \
oeD
[(J3±V(fi2 + 4y))/2
ifx is principal on G,
ifx isnonprincipalon G,
where (1 = X—^i, y = k—X if OED, and y = k — fj, ifQ$D.
Received 18 January 1990.
1980 Mathematics Subject Classification 05B10.
Bull. London Math. Soc. 22 (1990) 533-539
534
KA HIN LEUNG AND SIU LUN MA
LEMMA 1.2. Let G be an abelian group of order mn, N be a subgroup ofG of order
n, and E be a subset of G. Then E is an (m, n, k, X)-relative difference set in G relative
to N if, for any character x of G,
(
k
ifx is principal on G,
y/{k—Xri)
ifx is nonprincipal on G but principal on N,
•\/k
ifx is nonprincipal on N.
2. Local rings
Let R be a finite local ring with its maximal ideal / generated by a prime element
n. Our first goal is to construct partial difference sets in R x R. Before that, let us recall
some basic properties of R.
PROPOSITION 2.1. Let R, I and n be as defined above. Then
(i) R/IS Fpd, where p # 0 is the characteristic and d is a positive integer;
(ii) there exists a smallest integer s such that Is = (ns) = 0;
(iii) {/': 1 ^ i < s) is the set of all proper ideals in R, and \I*\ = /7<t(s~<>.
Let U = R\I be the set of units in R. It is easy to see that there exists
{xx,x2,...,xd}
a U such that the projection forms a basis of R/I over its prime field
Fp. Also, it follows from the above proposition that/) = nru, where r is an integer and
u e U. As an additive group, R is clearly an abelian p-gwup. In order to decompose
it into a direct summand of cyclic groups, we first prove the following.
LEMMA 2.2. Suppose s = qr + s', where both q and s' are integers with 0 ^s' < r.
Then for any 1 ^ / ^ d, 0 < j ^ r— 1,
. pq
otherwise.
Proof. The proof is obvious.
PROPOSITION
2.3. Let q and s' be as defined in Lemma 2.1. Then as an additive
group,
*=n n i/p^z x fi n z/^z}-0
(-1
1-S' i - 1
Proof. It suffices to show that R = ©f_! ©J'o** 7 ^By comparing the number of elements on both sides, it is enough to show
i-l
?-0
(Here we obtain an upper bound for the number of elements on the right by using
Lemma 2.2.)
Observe that for any unit ueU, there exist positive integers ai such that
u—Ydi~iatxie^Since nr=pu', where u'eU, we see easily that for any nonzero
xe/', there exists X ' G G such that X—X'EIi+1. By repeating this process, we prove that
xeG.
CONSTRUCTIONS OF PARTIAL DIFFERENCE SETS
535
Next, we shall show that the local rings described above exist. To construct R, we
need some well-known facts about p-adic fields.
Let Q p be the/>adic completion of Q and Z p be the set of/>adic integers, that is,
It is wen< known that pZp is the unique maximal
~%-v'-~ {Yii^oatPtm-^ ^ ai <P~^)ideal in Zp and that lp/(p) ^ Fp. (For details, see Chapter 4 of [2].)
Now let us consider a finite extension K of Qp. In K, we denote the integral closure
of Zp by R'. It is well known that R' is also a local ring with its maximal ideal
generated by an element n. Moreover, R'/nR' is a finite extension of Fp and p = nru,
where « is a unit in R' [2, Chapter 7]. So in order to construct the desired ring R, we
simply define R:= R'/nsR'. To sum up, we have the following.
PROPOSITION
2.4.
Let s, d and r be fixed positive integers. Then for any prime
number p, there exists a finite local ring R such that
(i) its maximal ideal 7 is generated by an element n ;
(ii) 7T8"1 # 0, n8 = 0 and p = nru, where u is a unit ;
(iii) R/I is a finite field of order pd.
From now on, we shall assume R, s, r, d and p to be as described in the above
proposition. Let T be a character on R such that T is nonprincipal on 7s"1. It is easy
to see that for any aeR, if we define ra(x) = r(ax) for all xeR, then xa is also a
character. Note that as r is nonprincipal on 7*"1, then xa — xb if and only if a = b.
Hence \{xa:aeR}\ = \R\ = the number of distinct characters on R. It follows that
every character is of the form TO, where aeR. Before we go on, let us observe some
properties of T.
LEMMA 2.5. For any 0 ^ i ^ s—\, Y^yeiiX(y)
/ * * # ( ) , then Zyeiiriyx) = 0.
=
®- I*1 particular, for any xeR, if
Proof This follows easily from the fact that r is nonprincipal on 7 s " 1 and 7* is
a group containing 7s"1.
LEMMA 2.6.
For any c$Is~x, £ M 6 ( / T(MC) = 0.
Proof. It suffices to show that
£ U 6 O + / T(WC)
= 0 for any ae U. Observe that
£ r(uc) = x(ac) £ x(xc).
uea+I
xel
s x
As c^I ~ , Ic is a nonzero ideal. Therefore by Lemma 2.5 above,
3. Partial difference sets
Let R be as defined in Proposition 2.4. In this section, our objective is to construct
partial difference sets in the additive group RxR. Our construction is described as
follows.
536
KA HIN LEUNG AND SIU LUN MA
3.1. Let 0 be a mapping from R to R such that <p(nru) = nru~l for
r = 0,1,2,..., s — 1 and UEU. Also, let f be a mapping from R to {0,1} such that
THEOREM
ld
e,
(1)
we/"" 1
where e is a constant. Then
is a partial difference set in the additive group of RxR with parameters
v
= p**, k = ep(8-1)d(psd -
\)+psdM,
X=p
REMARK. Note that uD = D for any ue U. In particular, we have —D = D and
hence D is reversible.
Let T be as defined in Section 2. In order to prove that D defined above is a partial
difference set, we need to know all the characters on R x R. The next lemma tells us
that any character on R x R is ' basically' induced by T.
LEMMA
(x,y)eRxR,
3.2. For any character x on RxR,
x(x,y) = x(ax + by).
there exist a,beR
such that for all
Proof. Let r1 and T2 be the restrictions of / on R x {0} and {0} x R respectively.
As observed in Section 2, there exist a, b eR such that for any x,yeR, T ^ O ) = Ta(x)
and T2(0,J>) = rb(y). Hence for any (x,y)eR x R, x(x,y) = ra(x)rb(y) = z(ax + by).
It follows from Lemmas 1.1 and 3.2 that D is a partial difference set with the given
parameters if and only if for any a,beR,
ep^d
or p8d-ep^d
otherwise.
<*>
F o r convenience, we define the following.
DEFINITION
3.3. F o r any z e R , we say that nr || z if z = nru for some unit u.
Let us n o w assume n m || b with 0 ^ m ^ s. T o prove Equation (2), we first prove
the following lemmas.
LEMMA
3.4. Let xeR. Ifnn\\ x, then
>
if 0 ^ n < m = s.
Proof. We first deal with the case when m <n^s. Note that p-™-1 ^ 0 and that
for any weR, </>(x)(w + r-m-1) = {<f>(x)w}. Therefore
= r(bw)M(x)w)
£
1,6/"~m-1
= 0 by Lemma 2.5.
x{by)
CONSTRUCTIONS OF PARTIAL DIFFERENCE SETS
537
It follows that ljyeRA</>(x)y)r(ax + by) = 0.
Next, we assume 0 ^ n < m. Observe that in this case {r(bw)} = r(b(w + Z""""1))
for any weR. Hence
£
A</>(x)y)T(by) = r(bw)
=
£
A</>(x) w + </>{x) y)
T(by)ep-a.
E
It follows that
yeR
yeR
Now if m = s, then the right-hand side becomes ep(s~1)d, whereas if m < s, then by
Lemma 2.5, the right-hand side is 0.
Let us remind readers that our goal is to prove Equation (2). In view of the above
lemma, we need to consider the case when nm || x. For convenience, let us define
A(m):={(x,y)eRxR:nm\\x}
and A(m, w):= {(x,y)€A(m):ax + by = w}.
LEMMA 3.5.
I — ep(s~1)d
Proof
becomes
ifm<s,
ifm = s.
By Lemma 2.6, we see easily that the left-hand side of the above equation
£
wel°
1
£ A<P(x)y)r(w).
(x,y)eA(m,w)
If w = 0, then </>(x)y = —<p(a)b. Hence
I if m < s,
(x,y)eA(m,0)
/KrX
'""
l/^'W)
ifw = 5.
(Note that we have used the fact that \R\ = psd and |/ r \/ r+1 ||/*|/|/ r | = psd-p{8-1)d.)
If wE/'^fO}, then <j){x)y = fiifya + xvb^XQ1, where bQ and x0 are units such that
nmb0 = b and nmx0 = x. Hence
E
M(*)y)=p{8-1)d
E /(tel'
\{0)
Now the lemma follows from Lemma 2.5.
Proof of Theorem 3.1. If b = 0, that is, s = m, then by Lemmas 3.4 and 3.5,
I
«/><"»• T(OX)+/>"./(0)
z6fl\{0)
0)-ep(s-1)d
ep{s-1)d(psd-l)
ifa^O,
if a = 0.
538
KA HIN LEUNG AND SIU LUN MA
If b # 0, then
E E-/W*)jO*(« + W =PsdA-<t>(b)")-eP{8-1)dxeR yeR
Since/(z) e {0,1} for all zeR, our theorem follows.
Finally, we explicitly state the family of reversible difference sets obtained in
Theorem 3.1.
3.6. In Theorem 3.1, if p = 2 and e = 2d~x, then D is a (22sd,
2 -i±2 - ,2 - ±2s<l-1ydijFerence set in RxR with -D = D. Furthermore, if
R = Z/281, then D is a Dillon's difference set.
COROLLARY
2sd
s<i 1
28<i 2
4. Relative difference sets
In this section, we construct relative difference sets using the partial difference sets
obtained in Section 3. Let {A1,...,Apt} be a partition of R such that for any coset
a + I8~l in R, 1 ^ i; ^ p \ we have 1^4, n a + Z8"1! = pd~\ where t is a positive integer less
than d. Furthermore, we denote the respective characteristic functions by/ 1 5 / 2 ,... ,fpt.
Obviously, they satisfy the following:
(0 HwBi'-'fii2 + w)= pd~l for all z e R and i = 1,2,... ,pl; and
Let us define Di = {(x,y)eRxR:fi(<^(x)y)
= 1} for i = l,2,...,pl. Note that
\jfmlDt = RxR, and by the proof of Theorem 3.1, for any character / of R x R,
'l(psd-1)
ifx is principal on RxR,
l l
if/ is nonprincipal on RxR,
~
where cx is an element independent of ft in R.
4.1. Let K = {g15 g 2 ,..., gpt} be an abelian group of order pl. Then E =
Uf-i(go A )
relative difference set in G = KxRxR relative to N = Kx{Q}x{0},
with parameters
THEOREM
is a
m = p2sa,
n—p\
k — p2sd
and X = p2sd~l.
Proof. Let x be any character of G. If/ is principal on G, then it is obvious that
YugeEXg = \RxR\= p 2 l d . Suppose / is nonprincipal on G but principal on N. Note
that / ' = /\ R x R is nonprincipal on RxR. So
P"
I! xg = T, ( T, x'x) = P8d Yfi(c -)-^y d " £ = o.
geE
i-1
geD{
i-l
Suppose / is nonprincipal on N. If / ' is principal on RxR, then
L yp =
Ao
ffeE
nsd V
psd~t(nv.rs<i — 1)/ uV Act
ye, = x^nsd
/ i /f.(O)
A y /yz
At; +
i Z'
Ayg,,
OJ5
v
r
<-l
t-1
where ^(0) = 1. (Here, we use the fact that only one/ ( (0) is nonzero, which is an easy
consequence of (ii).) So | YjgesXgl = Psd- Similarly, the same answer is obtained when
/ ' is nonprincipal on RxR. Hence the theorem follows by Lemma 1.2.
CONSTRUCTIONS OF PARTIAL DIFFERENCE SETS
539
REMARK. It can be shown that the construction in Theorem 4.1 also works even
if K is nonabelian. However, the proof is more tedious.
The following is a family of reversible relative difference sets constructed.
COROLLARY 4.2. In Theorem 4.1, ifp = 2 and K is an elementary 2-group, then E
is a (228d,2t,22sd,228d-t)-relative difference set in G relative to N with -E = E.
References
1. A. T. BUTSON, 'Relations among generalized Hadamard matrices, relative difference sets and maximal
length linear recurring sequences', Canad. J. Math. 15 (1963) 42-48.
2. J. W. S. CASSELS, Local fields (Cambridge University Press, 1986).
3. J. F. DILLON, 'Difference sets in 2-groups', Proc. NSA Math. Sci. Meeting (1987) 165-172.
4. D. JUNGNICKEL, 'On automorphism groups of divisible designs', Canad. J. Math. 34 (1982) 257-297.
5. S. L. MA, 'Partial difference sets', Discrete Math. 52 (1984) 75-89.
6. S. L. MA, 'On association schemes, Schur rings, strongly regular graphs and partial difference sets', Ars
Combinatoria 27 (1989) 211-220.
Department of Mathematics
National University of Singapore
Singapore 0511